Algorithm 2. Computation of log-likelihood function.

[$L_p,{\{x_k^i,w_{p,k}^i\}}_{i=1}^{N_s}]:=\mathtt{LL}[L_p,{\{x_{k-N:k-1}^i,w_{p,k-1}^i\}}_{i=1}^{N_s},y_k$]

for ($i=1:N_s$) –Draw $x_k^i\sim q\left(x_k\right|x_{0:k-1}^i,y_{1:k})$ – Assign particle a weight: $$w_{p,k}^i=w_{p,k-1}^i{\displaystyle \frac{P_p\left(y_k\right|x_{k-N:k}^i,y_{k-1})P\left(x_k^i\right|x_{k-1}^i)}{\mathrm{q}\left(x_k^i\right|x_{0:k-1}^i,y_{1:k})}}$$ end for Compute the LL function: $$L_p:=L_p+log\left({\displaystyle \frac{1}{N_s}}\sum _{i=1}^{N_s}{P}_p\left(y_k\right|x_{k-N:k}^i,y_{k-1})\right)$$ for ($i=1:N_s$) –Normalize the importance weight: $w_{p,k}^i:=w_{p,k}^i/\mathtt{SUM}[{\{w_{p,k}^i\}}_{i=1}^{N_s}]$ end for Resample the drawn particles at each step:  $[{\{x_k^j,w_{p,k}^j\}}_{j=1}^{N_s}]:=\mathtt{RESAMPLE}[{\{x_k^i,w_{p,k}^i\}}_{i=1}^{N_s}]$